Pipes and Cisterns – MBA CUET PG – Notes & Practice Questions

Introduction

Inlet Pipe

• Fills the tank.

• Has positive rate.
  If it fills the tank in x hours,

   

  $$\text{Rate} = \frac{1}{x} \text{ tank per hour}$$

   

Outlet Pipe

• Empties the tank.

• Has negative rate.
  If it empties the tank in y hours,

   

  $$\text{Rate} = -\frac{1}{y}$$

  ​

Combined Work

$$\text{Net Rate} = \text{Sum of individual rates}$$

Standard Formulas

If two pipes A and B fill a tank in a and b hours

$$\text{Together Time} = \frac{ab}{a + b}$$

If A fills in a hours and B empties in b hours (b > a)

$$\text{Net Time} = \frac{ab}{b – a}$$

If outlet pipe empties faster (b < a)

→ Tank never fills.

Three-Pipe Combined Time

If A, B, C fill in a, b, c hours:

$$\text{Net Rate} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$

$$\text{Time} = \frac{1}{\text{Net Rate}}$$

If one is an outlet (say C empties in c hours):

$$\text{Net Rate} = \frac{1}{a} + \frac{1}{b} – \frac{1}{c}$$

Part Fill / Empty Problems

If pipe works for t hours

$$\text{Work done} = \text{Rate} \times t$$

Find leftover work and then compute time based on remaining pipes.

Efficiency Method

Assign tank capacity = LCM of times to simplify.

Example: A fills in 6 h, B in 8 h
→ Take capacity = 24 units
→ A = 4 units/h, B = 3 units/h
→ Together = 7 units/h
→ Time = 24/7

This method is fastest for exam.

Leakage Problems

If a tank fills in a hours but leakage causes actual fill time = b hours

Leak rate = difference between effective and original.

$$\text{Leak Rate} = \frac{1}{a} – \frac{1}{b}$$

​

Leak empties the tank in:

$$\frac{1}{\left(\frac{1}{a} – \frac{1}{b}\right)}$$

​

Useful for “leak at bottom” type questions.

Partial Tank Cases

If tank is half full and pipe fills in x hours:

Time to fill remainder =

$$x \times \frac{\text{required fraction}}{1}$$

​

General:
If required fraction = f
Time = x × f

Time Inverse Relation Trick

If rate increases by k%, time decreases by:

$$\frac{k}{100+k} \times 100\%$$

If rate decreases by k%, time increases by:

$$\frac{k}{100-k} \times 100\%$$

Used when changing pipe speed or adding extra pipes.

Short Tricks

Pipes in Ratio

If A, B, C’s filling times are in ratio a : b : c
→ Rates are inverse: 1/a : 1/b : 1/c

Used for speed.

When A works alone for t hours

Remaining fraction =

$$1 – \frac{t}{a}$$

​

Then compute finishing time with other pipes.

When inlet and outlet alternate (hourly)

Compute net volume each cycle.

Pipe faster/slower by m%

If pipe A is m% more efficient than B:

$$\frac{1}{a} = \frac{1+m/100}{b}$$

​

Good for comparison questions.

Time Doubles Due to Leak

If without leak time = x, with leak = y:

$$\text{Leak empties in} = \frac{xy}{y – x}$$

​

Very common shortcut.

Typical Structures Seen in Exams

Type 1: Direct fill/empty

A fills in 10h, B in 12h → find together.

Type 2: Inlet + Outlet

Two fill, one empties — find net.

Type 3: Leakage

Tank takes longer due to leak → find leak time.

Type 4: Alternate Hours

A, B alternate every hour → compute cycle work.

Type 5: Starts/Stops

Pipe A works alone for some time, then B joins.

Type 6: Fraction-Based

Tank ⅔ full, pipe works for x hours → time for full.

Type 7: Container Volume Given

Rates converted to liters/min.

Fast Solve Examples

Pattern 1: A fills in a hrs, B empties in b hrs

Effective time:

$$T = \frac{ab}{b-a}$$

​

Pattern 2: Fill time = x, leak causes actual time = y

Leak empty time:

$$L = \frac{xy}{y-x}$$

​

Pattern 3: Two pipes fill in a, b hrs. Third empties in c hrs

$$\text{Net Rate} = \frac{1}{a} + \frac{1}{b} – \frac{1}{c}$$

​

Pattern 4: Partially full tank

If tank is p% full:
Remaining fraction = (100 – p)/100.

High-Value Fractions to Remember

• 1/2 tank = 0.5

• 1/3 tank = 0.333

• 2/3 = 0.666

• 1/4 = 0.25

• 3/4 = 0.75

Useful for “pipe works for 2 hrs then stops” type problems.

Practice Questions

1. Pipe A fills a tank in 12 h and Pipe B fills in 18 h. An outlet C empties it in 36 h. If all three are opened, time to fill?
A) 10 h
B) 12 h
C) 15 h
D) 18 h

2. Pipe A can fill a tank in 20 h. Due to a leak, it takes 25 h. Leak alone can empty in:
A) 50 h
B) 75 h
C) 100 h
D) 125 h

3. A and B together fill a tank in 24 h. A alone fills in 40 h. B alone fills in:
A) 48 h
B) 60 h
C) 56 h
D) 72 h

4. A pipe can fill 3/5 of a tank in 6 h. Time to fill full tank?
A) 8 h
B) 9 h
C) 10 h
D) 12 h

5. Pipe A fills in 15 h. Pipe B empties in 30 h. If both opened with tank initially full, tank empties in:
A) 20 h
B) 30 h
C) 45 h
D) 60 h